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(1 point) Enter a 3 x 3 symmetric matrix A that has entries 211 = 2,...
(1 point) The inverse of the matrix 1 -2 A = - 1 1 1 -1 0 is 011 012 013 A-1 421 022 023 031 032 033 where 011 = 012 = 013 = 021 = 022 = A23 = 031 = 432 = 433 =
Assume a symmetric matrix 3 x 3 matrix has eigenvalues 2 and 3 and that the eigenspace of 2 is E2 = span -{80 Which of the following is an eigenvector for the eigenvalue 3? O -6 O Por O 37
(1 point) Suppose A is a 3 x 3 matrix with real entries that has a complex eigenvalue 2 - 5i with corresponding eigenvector 9+3i1 1 .Find another eigenvalue and 42 eigenvector for A. Eigenvalue Eigenvector-
-2, 1), and 4. A is a 2 x 2 matrix with real entries, N(A - 31) = N(A - 1) = c(1,2) for all parts of this problem. (a) (4 points) Is A symmetric? (b) (4 points) Write the solution to the system of differential equations u' (t) = Au(t) if 7(0) = (6,7). (c) (4 points) What is 5e^? Write your answer as a single matrix.
مل 3 (1 point) Suppose that a 2 x 2 matrix A has an eigenvalue 3 with corresponding eigenvector and an eigenvalue -1 with corresponding eigenvector Find an invertible matrix P and a diagonal matrix D so that A = PDP-1. Enter your answer as an equation of the form A = PDP-1. You must enter a number in every answer blank for the answer evaluator to work properly. 1-1
Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest.) 06 6 0 60-3 2 For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.) dim(x)
1. For each of the following symmetric matrices, find an orthogonal matrix P and diagonal matrix D such that PTAP = D. 0 1 (а) А — 1 0 1 -1 1 0 2 -2 (Ъ) А %— -2 -2 -4 -2 2 |3 0 7 0 5 0 7 0 3 (с) А %— 1. For each of the following symmetric matrices, find an orthogonal matrix P and diagonal matrix D such that PTAP = D. 0 1 (а)...
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What properties 4. can be guaranteed for the corresponding unit eigenvectors u, v, w? In terms of u, v, w describe the nullspace, left nullspace, (b) row space, and column space of A (c) Find a vector x that satisfies Ax v +w. Is x unique? Under what conditions on b does Ax = b have a solution? (d) (e) If u, v, w are...
2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7). 3. Let A be an n x...
(1 Consider the symmetric matrix A = 2 10 2 0 2 2 1. Answer the following questions. 2 3 (1) Find the eigenvalues , , and iz (2 <, <1z) of the matrix A and their corresponding eigenvectors. (2) Find the orthogonal matrix B and its inverse matrix B' that satisfy the following equation: (4 0 0 B-'AB = 0 0 lo o 2) (3) Suppose that the real vectors y and 9 satisfy the following relationship: Show that...